Given a reductive group G, we introduce a class of moduli problems of framed principal G-bundles on chains of projective lines. Their moduli stacks provide equivariant toroidal compactifications of G. All toric orbifolds are examples of this construction, as are the wonderful compactifications of adjoint groups of De Concini-Procesi. As an additional benefit, we show that every semi-simple group has a canonical orbifold compactification. We further indicate the connection with non-abelian symplectic cutting and the Losev-Manin spaces. This is joint work with Michael Thaddeus (Columbia U).

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